2 Answers
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I've observed a number of cyclocross races in public parks in the area I live. These typically include paved areas, grassy areas and some dirt. If there's been rain recently (or raining during the race), the grass along the race course looks like it's been turned to mud. If you look more closely, you can see that the individual blades of grass are all intact, they're just bent over and muddy. A few months later, you can't even tell where the race course was anymore.

I believe it depends on the type of grass. For well-established and regularly mowed grass, such as is common in parks and for playing sports on, the damage is temporary and cosmetic. The ruts settle out on their own.

It seems like areas of slope that cause the racers to dig in and slip around more can get more damage.

The best thing to do after the race is rake out (flatten out) the ruts, the grass will come back but you don't want ruts in your grass.

Course layout can help, try to limit turns on grass, rolling straight through is generally fine unless really wet and soft. We put on a race today and routed to take advantage of gravel paths in the park, open areas under evergreen trees (lots of pine needles) and zigged and zagged a path through the trees. We did have sections on grass but mostly kept to long straights.

Time of year can help too, last year, this same race was below zero (no snow) but frozen ground which hardened the grass areas and had no mud. We cannot schedule around weather, but if you are in an area that is warm in September but frozen in December, I'd pick December.